Kite's thoughts on 41edo Lattices: Difference between revisions
added lattices of intervals, some rewriting too |
m TallKite moved page 41edo Lattices to Kite's thoughts on 41edo Lattices: Editor VectorGraphics has repeatedly edited pages I've written about my own research and inserted misinformation. On discord he is openly hostile to me. This move is necessary to avoid a toxic work environment for me. |
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== Lattices == | == Lattices == | ||
=== The 5-limit (ya) Lattice === | === The 5-limit (ya) Lattice === | ||
This lattice uses [[Ups and | This lattice uses [[Ups and downs notation|ups and downs notation]] to name the [[41-edo|41-edo (aka 41-equal)]] notes: | ||
[[File:41equal lattice 5-limit.png|none|thumb|456x456px]] | [[File:41equal lattice 5-limit.png|none|thumb|456x456px]] | ||
The middle row is a chain of 5ths. Moving one step to the right (aka '''fifthwards''') adds a 5th, and one step to the left ('''fourthwards''') adds a 4th. Just like the 12-equal circle of 5ths, octave equivalence is assumed and each note represents an entire pitch class. Moving diagonally right-and-up (the 1:00 direction) adds a downmajor 3rd, 5/4. This '''yoward''' step adds prime 5. Moving '''guward''' left-and-down subtracts prime 5. (The terms yo and gu come from [[color notation]].) Since moving 5thwards/4thwards adds/subtracts prime 3, and the octave is prime 2, every octave-reduced 5-limit ratio appears exactly once in the lattice. Thus the lattice is a "map" of all possible 5-limit notes. | The middle row is a chain of 5ths. Moving one step to the right (aka '''fifthwards''') adds a 5th, and one step to the left ('''fourthwards''') adds a 4th. Just like the 12-equal circle of 5ths, octave equivalence is assumed and each note represents an entire pitch class. Moving diagonally right-and-up (the 1:00 direction) adds a downmajor 3rd, 5/4. This '''yoward''' step adds prime 5. Moving '''guward''' left-and-down subtracts prime 5. (The terms yo and gu come from [[color notation]].) Since moving 5thwards/4thwards adds/subtracts prime 3, and the octave is prime 2, every octave-reduced 5-limit ratio appears exactly once in the lattice. Thus the lattice is a "map" of all possible 5-limit notes. | ||
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Chord progressions can be mapped out on the lattice as a series of chord shapes. Often two adjacent chords have a common tone, and the progression "walks" around the lattice. If there are no common tones, the progression "jumps" from one area to another. Often there are several different places to jump. More on this later. | Chord progressions can be mapped out on the lattice as a series of chord shapes. Often two adjacent chords have a common tone, and the progression "walks" around the lattice. If there are no common tones, the progression "jumps" from one area to another. Often there are several different places to jump. More on this later. | ||
This lattice shows the notes as intervals, and shows how many edosteps each note is from the tonic D. | This lattice shows the notes as intervals, and shows how many [[Arrow|arrows]] (edosteps) each note is from the tonic D. | ||
[[File:The Kite Tuning lattices-1.png|none|thumb|499x499px]] | [[File:The Kite Tuning lattices-1.png|none|thumb|499x499px]] | ||
This lattice is smaller than the previous one, but it still extends far enough to include all 41 notes, except 16 and 25 (v4 and ^5). By coincidence, each number appears exactly once. | This lattice is smaller than the previous one, but it still extends far enough to include all 41 notes, except 16 and 25 (v4 and ^5). By coincidence, each number appears exactly once. | ||
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== Commas == | == Commas == | ||
A comma is a just intonation ratio that is small, less than (roughly) 50¢. In 41-edo aka 41-equal, a comma maps to a small number of | A comma is a just intonation ratio that is small, less than (roughly) 50¢. In 41-edo aka 41-equal, a comma maps to a small number of arrows, usually 0 or 1, occasionally 2. The technical term for a comma that maps to 0 arrows is '''vanishing''' comma. On the Kite guitar, mapping to 0, 1 or 2 arrows can be called a no-fret, half-fret or one-fret comma. | ||
=== 5-limit (ya) commas === | === 5-limit (ya) commas === | ||
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!yaza comma | !yaza comma | ||
!as a yazala comma | !as a yazala comma | ||
!as a yazalatha | !as a yazalatha comma | ||
|- | |- | ||
|red vCx | |red vCx | ||
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== Commas Part II == | == Commas Part II == | ||
The Kite guitar fretboard is made up of alternating rainbow zones and | The Kite guitar fretboard is made up of alternating rainbow zones and complex zones. Playing a Gu or Ru pump on the Kite guitar forces one to either shift a pitch or walk into the complex zone. Going into the complex zone makes the tonic drift sharp or flat by half a fret. But playing a Saruyo pump forces one to walk up the neck clear through the complex zone into the next rainbow zone. The tonic drifts flat and then sharp! Depending on the melody, this can result in the scale temporarily containing offperfect tonics, 4ths and 5ths, which can be disconcerting. | ||
We can call Saruyo a walk-once comma, and Ru and Gu walk-halfway commas. All no-fret commas are no-walk, walk-once, walk-twice, etc. All half-fret commas are walk-halfway, walk-one-and-a-half, etc. | We can call Saruyo a walk-once comma, and Ru and Gu walk-halfway commas. All no-fret commas are no-walk, walk-once, walk-twice, etc. All half-fret commas are walk-halfway, walk-one-and-a-half, etc. | ||